Foundations of Computational Mathematics最新文献_第2页

Tribute to Nick Higham 向尼克-海勒姆致敬

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-22 DOI: 10.1007/s10208-024-09680-9

Alan Edelman

引用次数: 0

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints 近端伽勒金：用于点式约束的结构保留有限元方法

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-20 DOI: 10.1007/s10208-024-09681-8

Brendan Keith, Thomas M. Surowiec

{"title":"Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints","authors":"Brendan Keith, Thomas M. Surowiec","doi":"10.1007/s10208-024-09681-8","DOIUrl":"https://doi.org/10.1007/s10208-024-09681-8","url":null,"abstract":"The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"99 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Classification of Finite Groups: Recent Developements and Open Problems 有限群的分类：最新发展和未决问题

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-12 DOI: 10.1007/s10208-024-09688-1

Bettina Eick

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Quantitative Convergence of a Discretization of Dynamic Optimal Transport Using the Dual Formulation 使用二元公式对动态优化运输进行离散化的定量收敛

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-11 DOI: 10.1007/s10208-024-09686-3

Sadashige Ishida, Hugo Lavenant

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Computing the Noncommutative Inner Rank by Means of Operator-Valued Free Probability Theory 利用算子值自由概率论计算非交换内等级

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-11 DOI: 10.1007/s10208-024-09684-5

Johannes Hoffmann, Tobias Mai, Roland Speicher

引用次数: 0

Gabor Phase Retrieval via Semidefinite Programming 通过半定量编程实现 Gabor 相位检索

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-11-07 DOI: 10.1007/s10208-024-09683-6

Philippe Jaming, Martin Rathmair

{"title":"Gabor Phase Retrieval via Semidefinite Programming","authors":"Philippe Jaming, Martin Rathmair","doi":"10.1007/s10208-024-09683-6","DOIUrl":"https://doi.org/10.1007/s10208-024-09683-6","url":null,"abstract":"We consider the problem of reconstructing a function (fin L^2({mathbb R})) given phase-less samples of its Gabor transform, which is defined by $$begin{aligned} {mathcal {G}}f(x,y) :=2^{frac{1}{4}} int _{mathbb R}f(t) e^{-pi (t-x)^2} e^{-2pi i y t},text{ d }t,quad (x,y)in {mathbb R}^2. end{aligned}$$More precisely, given sampling positions (Omega subseteq {mathbb R}^2) the task is to reconstruct f (up to global phase) from measurements ({|{mathcal {G}}f(omega )|: ,omega in Omega }). This non-linear inverse problem is known to suffer from severe ill-posedness. As for any other phase retrieval problem, constructive recovery is a notoriously delicate affair due to the lack of convexity. One of the fundamental insights in this line of research is that the connectivity of the measurements is both necessary and sufficient for reconstruction of phase information to be theoretically possible. In this article we propose a reconstruction algorithm which is based on solving two convex problems and, as such, amenable to numerical analysis. We show, empirically as well as analytically, that the scheme accurately reconstructs from noisy data within the connected regime. Moreover, to emphasize the practicability of the algorithm we argue that both convex problems can actually be reformulated as semi-definite programs for which efficient solvers are readily available. The approach is based on ideas from complex analysis, Gabor frame theory as well as matrix completion. As a byproduct, we also obtain improved truncation error for Gabor expensions with Gaussian generators.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"61 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142597481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Theory of the NEPv Approach for Optimization on the Stiefel Manifold 斯蒂费尔曼菲尔德上优化的 NEPv 方法理论

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-10-31 DOI: 10.1007/s10208-024-09687-2

Ren-Cang Li

{"title":"A Theory of the NEPv Approach for Optimization on the Stiefel Manifold","authors":"Ren-Cang Li","doi":"10.1007/s10208-024-09687-2","DOIUrl":"https://doi.org/10.1007/s10208-024-09687-2","url":null,"abstract":"The NEPv approach has been increasingly used lately for optimization on the Stiefel manifold arising from machine learning. General speaking, the approach first turns the first order optimality condition into a nonlinear eigenvalue problem with eigenvector dependency (NEPv) and then solve the nonlinear problem via some variations of the self-consistent-field (SCF) iteration. The difficulty, however, lies in designing a proper SCF iteration so that a maximizer is found at the end. Currently, each use of the approach is very much individualized, especially in its convergence analysis phase to show that the approach does work or otherwise. Related, the NPDo approach is recently proposed for the sum of coupled traces and it seeks to turn the first order optimality condition into a nonlinear polar decomposition with orthogonal factor dependency (NPDo). In this paper, two unifying frameworks are established, one for each approach. Each framework is built upon a basic assumption, under which globally convergence to a stationary point is guaranteed and during the SCF iterative process that leads to the stationary point, the objective function increases monotonically. Also the notion of atomic function for each approach is proposed, and the atomic functions include commonly used matrix traces of linear and quadratic forms as special ones. It is shown that the basic assumptions of the approaches are satisfied by their respective atomic functions and, more importantly, by convex compositions of their respective atomic functions. Together they provide a large collection of objectives for which either one of approaches or both are guaranteed to work, respectively.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"67 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142562122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Explicit A Posteriori Error Representation for Variational Problems and Application to TV-Minimization 变量问题的显式后验误差表示法及其在电视最小化中的应用

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-10-18 DOI: 10.1007/s10208-024-09676-5

Sören Bartels, Alex Kaltenbach

引用次数: 0

The Gromov–Wasserstein Distance Between Spheres 球体间的格罗莫夫-瓦瑟施泰因距离

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-09-16 DOI: 10.1007/s10208-024-09678-3

Shreya Arya, Arnab Auddy, Ranthony A. Clark, Sunhyuk Lim, Facundo Mémoli, Daniel Packer

{"title":"The Gromov–Wasserstein Distance Between Spheres","authors":"Shreya Arya, Arnab Auddy, Ranthony A. Clark, Sunhyuk Lim, Facundo Mémoli, Daniel Packer","doi":"10.1007/s10208-024-09678-3","DOIUrl":"https://doi.org/10.1007/s10208-024-09678-3","url":null,"abstract":"The Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov–Wasserstein distance and the assessment of the approximation quality of computational techniques designed to estimate the Gromov–Wasserstein distance, we determine the precise value of a certain variant of the Gromov–Wasserstein distance between unit spheres of different dimensions. Indeed, we consider a two-parameter family ({d_{{{text {GW}}}p,q}}_{p,q=1}^{infty }) of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we are able to determine the exact value of the distance (d_{{{text {GW}}}4,2}) between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"13 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142235260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Unbiasing Hamiltonian Monte Carlo Algorithms for a General Hamiltonian Function 一般哈密顿函数的无偏哈密顿蒙特卡洛算法

IF 3 1区数学

Foundations of Computational Mathematics Pub Date : 2024-09-16 DOI: 10.1007/s10208-024-09677-4

T. Lelièvre, R. Santet, G. Stoltz

{"title":"Unbiasing Hamiltonian Monte Carlo Algorithms for a General Hamiltonian Function","authors":"T. Lelièvre, R. Santet, G. Stoltz","doi":"10.1007/s10208-024-09677-4","DOIUrl":"https://doi.org/10.1007/s10208-024-09677-4","url":null,"abstract":"Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo method that allows to sample high dimensional probability measures. It relies on the integration of the Hamiltonian dynamics to propose a move which is then accepted or rejected thanks to a Metropolis procedure. Unbiased sampling is guaranteed by the preservation by the numerical integrators of two key properties of the Hamiltonian dynamics: volume-preservation and reversibility up to momentum reversal. For separable Hamiltonian functions, some standard explicit numerical schemes, such as the Störmer–Verlet integrator, satisfy these properties. However, for numerical or physical reasons, one may consider a Hamiltonian function which is nonseparable, in which case the standard numerical schemes which preserve the volume and satisfy reversibility up to momentum reversal are implicit. When implemented in practice, such implicit schemes may admit many solutions or none, especially when the timestep is too large. We show here how to enforce the numerical reversibility, and thus unbiasedness, of HMC schemes in this context by introducing a reversibility check. In addition, for some specific forms of the Hamiltonian function, we discuss the consistency of these HMC schemes with some Langevin dynamics, and show in particular that our algorithm yields an efficient discretization of the metropolized overdamped Langevin dynamics with position-dependent diffusion coefficients. Numerical results illustrate the relevance of the reversibility check on simple problems.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"185 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142235261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0